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A258760
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Decimal expansion of Ls_4(Pi/3), the value of the 4th basic generalized log-sine integral at Pi/3.
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4
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6, 0, 0, 9, 4, 9, 7, 5, 4, 9, 8, 1, 8, 8, 8, 8, 9, 1, 6, 2, 0, 4, 7, 8, 8, 7, 0, 6, 2, 0, 3, 2, 7, 0, 7, 4, 0, 5, 9, 6, 9, 6, 3, 2, 9, 7, 4, 3, 9, 5, 6, 8, 4, 1, 8, 8, 3, 6, 0, 6, 3, 9, 2, 6, 7, 5, 1, 5, 1, 0, 0, 4, 2, 0, 0, 2, 8, 0, 2, 2, 5, 2, 6, 8, 7, 6, 2, 3, 8, 6, 2, 3, 6, 9, 0, 5, 6, 6, 3, 5, 9, 3, 0, 5, 3
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OFFSET
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1,1
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LINKS
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FORMULA
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-Integral_{0..Pi/3} log(2*sin(x/2))^3 dx = (1/2)*Pi*zeta(3) + (9/4)*im( PolyLog(4, (-1)^(1/3)) - PolyLog(4, -(-1)^(2/3))).
Also equals 6 * 5F4(1/2,1/2,1/2,1/2,1/2; 3/2,3/2,3/2,3/2; 1/4) (with 5F4 the hypergeometric function).
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EXAMPLE
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6.00949754981888891620478870620327074059696329743956841883606392675151...
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MATHEMATICA
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RealDigits[(1/2)*Pi*Zeta[3] + (9/4)*Im[ PolyLog[4, (-1)^(1/3)] - PolyLog[4, -(-1)^(2/3)]], 10, 105] // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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